At the same time that Hairer introduced his theory of regularity structures, Gubinelli, Imkeller and Perkowski developed paracontrolled calculus as an alternative playground where to study a number of singular, classically ill-posed, stochastic partial differential equations, such as the or -dimensional parabolic Anderson model equation (PAM)
the equation of stochastic quantization
or the one dimensional KPZ equation
to name but a few examples. In each of these equations, the letter stands for a space or time/space white noise who is so irregular that we do not expect any solution of the equation to be regular enough for the nonlinear terms, or the product , in the equations to make sense on the sole basis of the regularizing properties of the heat semigroup. Like Hairer’s theory of regularity structures, paracontrolled calculus provides a setting where one can make sense of such a priori ill-defined products, and finally give some meaning and solve some singular partial differential equations. We present here an overview of paracontrolled calculus, from its initial form to its recent extensions.
@article{JEDP_2016____A1_0, author = {Bailleul, Isma\"el}, title = {Paracontrolled calculus}, journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles}, note = {talk:1}, pages = {1--11}, publisher = {Groupement de recherche 2434 du CNRS}, year = {2016}, doi = {10.5802/jedp.642}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jedp.642/} }
Bailleul, Ismaël. Paracontrolled calculus. Journées équations aux dérivées partielles (2016), Talk no. 1, 11 p. doi : 10.5802/jedp.642. http://www.numdam.org/articles/10.5802/jedp.642/
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